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Graph Logics and Relations on Words

We investigate some basic questions about the interaction of regular and rational relations on words.

Abstract

We investigate some basic questions about the interaction of regular and rational relations on words. The primary motivation comes from the study of logics for querying graph topology, which have recently found numerous applications. Such logics use conditions on paths expressed by regular languages and relations, but they often need to be extended by rational relations such as subword (factor) or subsequence. Evaluating formulae in such extended graph logics boils down to checking nonemptiness of the intersection of rational relations with regular or recognizable relations.

We prove that for several basic and commonly used rational relations, the intersection problem with regular relations is either undecidable (e.g., for subword or suffix, and some generalizations), or decidable with non-multiply-recursive complexity (e.g., for subsequence and its generalizations). These results are used to rule out many classes of graph logics that freely combine regular and rational relations, as well as to provide the simplest problem related to verifying lossy channel systems that has non-multiply-recursive complexity. We then prove a dichotomy result for logics combining regular conditions on individual paths and rational relations on paths, by showing that the syntactic form of formulae classifies them into either efficiently checkable or undecidable cases. We also give examples of rational relations for which such logics are decidable even without syntactic restrictions.

This is joint work with Pablo Barcelo and Diego Figueira.

Biography

Leonid Libkin is Professor of Foundations of Data Management in the School of Informatics at the University of Edinburgh. He was previously a Professor at the University of Toronto and a member of research staff at Bell Laboratories in Murray Hill. He received his PhD from the University of Pennsylvania in 1994. His main research interests are in the areas of data management and applications of logic in computer science. He has written four books and over 150 technical papers. He was the recipient of a Marie Curie Chair Award from the EU in 2006, a Premier's Research Excellence Award in 2001, and won four best paper awards. He has chaired programme committees of major database conferences (ACM PODS, ICDT) and was the conference chair of the 2010 Federated Logic Conference. He has given a dozen invited conference talks and has served on multiple program committees and editorial boards. He is a fellow of the Royal Society of Edinburgh.

Note

This lecture is organized by the Vienna Center for Logic and Algorithms (VCLA) in cooperation with the Kurt Goedel Society.